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8. Suppose that there are two producers of prokrypton-B, a rare mineral capable of incapacitating aliens from distant planets. Governments across the globe are interested in purchasing defensive stores of the rare mineral. The market demand for prokrypton-B is given by P= 200 –0.2Q,where Qis the collective output of the industry, in tons; q Ais the output of Firm A; and q Bis the output of Firm B. Assume that each firm produces prokrypton-B at a marginal cost of $10 per ton.a. Derive each firm’s reaction function.b. Solve for the equilibrium quantity of each firm, as well as industry output.c. Solve for the market price of prokrypton-B.d. Solve for the profit earned by each firm.e. Suppose that a labor dispute drives Firm B’s cost to $12 per ton.i. What happens to the output of each firm as a result?ii. What happens to the market price of prokrypton-B?iii. What happens to the profit of each firm?8. a. The inverse demand isP= 200 –0.2(q A+ q The residual marginal revenue for Firm A isMR A= 200 –0.4q A–0.2Therefore, the reaction function for Firm A isMR A= 200 –0.4 q A–0.2 q B= 10 = q A= 475 –0.5q The residual marginal revenue for Firm B isMR B= 200 –0.2q A–0.4 Therefore, the reaction function for Firm B isMR B= 200 –0.2q A–0.4q B= 10 = q B= 475 –0.5q b. The quantity produced by Firm A isq A= 475 –0.5q B= 475 –0.5(475 –0.5 q A) = 237.5 + 025 q ASimilarly, Firm B produces 316.67 tons of prokrypton-B. Thus, the equilibrium quantity for the indus-try isQ= q B+ q A= 633.33B)q BMCBq BMCA= 316.67Goolsbee1e_Solutions_Manual_Ch11.indd 151Goolsbee1e_Solutions_Manual_Ch11.indd 15111/15/12 3:09 PM11/15/12 3:09 PM
152Part 3 Markets and PricesSolutionc. The market price of prokrypton-B isP= 200 –0.2Q= 200 –0.2 × 633.33 = $73.33d. The profit for Firm A isTR–TC= ($73.33 –$10) × 316.67 = $20,055.56Firm B receives the same profit.e. i. The reaction function for Firm A is unchanged. The new profit-maximizing condition for Firm B isMR B= 200 –0.2q A–0.4 q B= 12 = q B= 470 –0.5 The quantity produced by Firm A isq A= 475 –0.5q B= 475 –0.5(470 –0.5 q A) = 240 + 0.25 = 320Firm B producesq B= 470 –0.5 q A= 310Hence, the output of Firm A increases and the output of Firm B decreases.ii. The market price for prokrypton-B isP= 200 –0.2Q= 200 –0.2 × (320 + 310) = $74The market price for prokrypton-B increases.iii. The profit for Firm A isTR–TC= ($74 –$10) × 320 = $20,480Firm B receivesTR–TC= ($74 –$12) × 310 = $19,220Firm A increases its profit, whereas Firm B decreases its profit. MCq Aq A